1 / 6

Math 80A Section 3.5 Relations and Functions

Math 80A Section 3.5 Relations and Functions. A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, then x corresponds to y, or y depends on x. Example:.

oliver-russell
Download Presentation

Math 80A Section 3.5 Relations and Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Math 80A Section 3.5 Relations and Functions A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, then x corresponds to y, or y depends on x. Example: The set of x-coordinates {2,3,4,4,5,6,6,7} corresponds to the set of y coordinates {60,70,70,80,85,85,95,90} The set of distinct x-coordinates is called the domain of the relation. This is the set of all possible x values specified for a given relation. The set of all distinct y values corresponding to the x-coordinates is called the range. In the example above, Domain = {2,3,4,5,6,7} Range = {60,70,80,85,95,90}

  2. FUNCTIONS Input x Function f(x) Output y • A function, f, is like a machine that receives as input a number, x, from the domain, manipulates it, and outputs the value, y. • The function is simply the process that x goes through to become y. This “machine” has 2 restrictions: • It only accepts numbers from the domain of the function. • For each input, there is exactly one output (which may be repeated for different inputs). “OFFICIAL” DEFINITION OF A FUNCTION: Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X, exactly one element of Y. However, an element of Y may have more than one elements of x associated with it. That is, for each ordered pair (x,y), there is exactly one y value for each x, but there may be multiple x values for each y. The variable x is called the independent variable (also sometimes called the argument of the function), and the variable y is called dependent variable (also sometimes called the image of the function.) Analogy: In the x-y “relation”-ship, the x’s are the wives and the y’s are the husbands. A husband is allowed to have two or more wives, but each wife(x) is only allowed 1 husband(y). This relation is not a function because there are two different y-coordinates for the x-coordinate, 4, and also for the x-coordinate, 6.

  3. Example: Given the relation {(-2,-3),(2,3),(-1,2),(-3,4),(3,4)} Is this relation a function?_______________ (does any one x-value have more than one corresponding y value?) What is the domain? -3,-2, -1, 2, 3 What is the range? -3, 2, 3, 4 Example: Given the relation {(-4,4),(-2,2),(0,0),(-2,-2)} Is this relation a function?_______________ (does any one x-value have more than one corresponding y value?) What is the domain?___________________ What is the range?______________________

  4. A relation is not a function of x if there is more than one corresponding y-value for any x-value. To be a function of x, put the relation in the form y = ___ and there should be only 1 possible solution for any one x-value. Example 6 p. 202 Are these functions of x? x = y2 y can be or Not a function of x. y = 2x There is only 1 possibility for any x. So YES. it is a function of x. x = |y| Solve for y. x = y or x = -y y = x or y = -x Two possibilities, therefore NOT A FUNCTION.

  5. VERTICAL LINE TEST A graph is the graph of a funciton if and only if there is no vertical line that crosses the graph more than once.

  6. Example Consider the equation for the line segment: y = 2x – 5, where the domain is {x|1 ≤ x ≤ 6} Is this equation a function? Notice that for any x, you can only get one answer for y. (E.g. when x =1, then y = 2(1) – 5= -3.) Therefore the equation is a function. Functional notation for this equation would be f(x) = 2x – 5 Just replace the y with f(x). Note: This is stated “f of x”, It does not mean f times x, though it looks like that. What is the range? Since this is a straight line, we need only check y values at endpoints of domain. The y values do not fluctuate between these endpoints. The y values are also called function values, so they are often referred to as f(x), which means the value of the function at x (not f times x). The endpoints of the domain are 1 and 6. f(1) = the value of the function (what is y?) when x = 2 = 2(1) – 5 = -3 f(6) = the value of the function (what is y?) when x = 6 = 2(6) – 5 = 7 So the range is {y|-3 ≤ y ≤ 7} Example Find the range of the function given by the equation f(x) = -3x + 2 if the domain is {-4,-2,0,2,4}. Recall domain means possible x-values. The ordered pairs that belong to this function are graphed to the right. They are: {(-4,14), (-2,8), (0,2),(2,-4),(4,-10)} Range means possible y values, so Range = {-10,-4,2,8,14} This figure is a line segment with endpoints (1,-3) and (6,7).

More Related